Problem
Problem 1.78 Let $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ be given by $f(x, y, z)=x^{2}+y^{2}-1$.
(i) Prove that $C=f^{-1}(0)$ is an embedded 2-submanifold of $\mathbb{R}^{3}$.
(ii) Prove that a vector
$$
v=\left(a \frac{\partial}{\partial x}+b \frac{\partial}{\partial y}+c \frac{\partial}{\partial z}\right)_{(0,1,1)}
$$
is tangent to $C$ if and only if $b=0$.
(iii) If $j: S^{1} \rightarrow \mathbb{R}^{2}$ is the inclusion map, prove that $j \times \mathrm{id}_{\mathbb{R}}: S^{1} \times \mathbb{R} \rightarrow \mathbb{R}^{3}$ induces a diffeomorphism from $S^{1} \times \mathbb{R}$ to $C$.